Article
Reference
Testing a Bell inequality in multipair scenarios
BANCAL, Jean-Daniel, et al.
Abstract
To date, most efforts to demonstrate quantum nonlocality have concentrated on systems of two (or very few) particles. It is, however, difficult in many experiments to address individual particles, making it hard to highlight the presence of nonlocality. We show how a natural setup with no access to individual particles allows one to violate the Clauser-Horne-Shimony-Holt inequality with many pairs, including in our analysis effects of noise and losses. We discuss the case of distinguishable and indistinguishable particles. Finally, a comparison of these two situations provides insight into the complex relation between entanglement and nonlocality.
BANCAL, Jean-Daniel, et al . Testing a Bell inequality in multipair scenarios. Physical Review.
A , 2008, vol. 78, no. 6, p. 1-8
DOI : 10.1103/PhysRevA.78.062110
Available at:
http://archive-ouverte.unige.ch/unige:1120
Disclaimer: layout of this document may differ from the published version.
1 / 1
Jean-Daniel Bancal1, Cyril Branciard1, Nicolas Brunner1, Nicolas Gisin1, Sandu Popescu2,3, and Christoph Simon1
1Group of Applied Physics, University of Geneva, 20 rue de l’Ecole-de-M´edecine, CH-1211 Geneva 4, Switzerland
2H.H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL, U.K.
3Hewlett-Packard Laboratories, Stoke Gifford, Bristol BS12 6QZ, U.K.
(Dated: November 11, 2008)
To date, most efforts to demonstrate quantum nonlocality have concentrated on systems of two (or very few) particles. It is however difficult in many experiments to address individual particles, making it hard to highlight the presence of nonlocality. We show how a natural setup with no access to individual particles allows one to violate the CHSH inequality with many pairs, including in our analysis effects of noise and losses. We discuss the case of distinguishable and indistinguishable particles. Finally, a comparison of these two situations provides new insight into the complex relation between entanglement and nonlocality.
PACS numbers: 03.65.Ud
I. INTRODUCTION
Entanglement is the resource that allows one to establish quantum non-local correlations [1]. These correlations have been the center of a wide interest, because of their fascinating nature, and of their impressive power for processing informa- tion. Experimentally, quantum non-locality is demonstrated in so-called Bell experiments, which have to date all confirmed the quantum predictions [2].
Most theoretical works on Bell experiments and Bell in- equalities have focused on the case where the source emits a single entangled pair of particles at a time. Indeed, this is the simplest situation to study. From the experimental point of view, most experiments have been designed in order to match this theoretical model. For example, in photonic ex- periments, the source, usually based on parametric down con- version (PDC), is set in the weak regime; i.e. when the source emits something, it is most likely a single pair of entangled photons.
However, there are experimental situations, such as in many-body systems, where producing single entangled pairs is rather difficult. For instance, in Ref. [3] many entangled pairs ('104) of ultracold atoms have been created but cannot be addressed individually. So, while entanglement has defi- nitely been created in this system, one still lacks an efficient method for demonstrating its quantum non-locality through the violation of some Bell inequality. The goal of the present paper is to discuss techniques for testing Bell inequalities in such multi-pair scenarios, where the particles on Alice’s and Bob’s side cannot be individually addressed, and must there- fore be measured globally (see Fig. 1). What we mean here by global measurements is that each particle is submitted to the same measurement. Note that the case of more general measurements (collective measurements on all particles) has been considered in Ref. [4].
Basically one should distinguish two cases: independent pairs and indistinguishable pairs. In the first case, the pairs are created independently, but cannot be addressed individually;
therefore they must be measured globally (both on Alice’s and on Bob’s side). During this global measurement, the classical information about the pairing is lost: there is no way to tell which particle is entangled with which. The corresponding
loss of entanglement has been derived in Ref. [5]. In the second case, the pairs are indistinguishable; so in some sense the information about the pairing is here lost in a coherent way.
Reidet al.[6] have considered the case of indistinguishable pairs (with global measurement) in optics. More specifically, these authors, extending on a previous work of Drummond [7], showed how Bell inequalities can be tested (and violated) when many pairs are created via PDC. In this case the pairs are indistinguishable because of the process of stimulated emis- sion. In Ref. [8], Joneset al. have considered a related sce- nario; there entangled pairs are delivered via an inept delivery service, but at the end only a single pair is measured. Also considering multi-particle entanglement in such scenario is an interesting problem : see for example [9, 10].
In this paper we will study the violation of Bell inequalities in a general multi-pair scenario. We start by treating the case of independent pairs (Section II). We argue that the resistance to noise is here the relevant measure of non-locality, evalu- ating it. The consequences of particle losses are also inves- tigated. Next we move to the case of indistinguishable pairs (Section III), after a brief review of the results of Ref. [6], we present an analysis of the influence of noise and losses in this case. In Section IV, we compare the entanglement and non-locality in both cases. This leads us to a surprising result:
while the state of indistinguishable pairs contains more entan- glement than the state of independent pairs (after the classi- cal mixing), the latest appears to be more non-local. In other words, the incoherent loss of information provides more non- locality, but less entanglement, than the coherent loss of in- formation (indistinguishable pairs). This provides a novel ex- ample (here in the case of multi-pairs) of the complex relation between entanglement and non-locality. Finally we provide some experimental perspectives (Section V) and conclusions.
II. INDEPENDENT PAIRS
We consider a source emittingM entangled pairs, each of them being in the same entangled two-qubit stateρ. Thus the global state is
arXiv:0810.0942v2 [quant-ph] 11 Nov 2008
2
FIG. 1: Setup : a source producesM independent pairs (or equiv- alently M independant sources each produce a pair), the pairing be- tween Alice’s and Bob’s particles is lost during their transmission, and each party measures all their incoming particles in the same ba- sis. The total numbern+(−)of particles detected in the+(−)out- come is tallied on both sides.
ρM =ρ⊗M =ρ⊗ρ⊗ · · · ⊗ρ
| {z }
Mtimes
. (1)
Each pair being independent, Alice and Bob receiveM un- correlated particles. Since Alice and Bob are unable to ad- dress single particles in their ensemble, they perform a global measurement on their M particles, i.e. all M particles are measured in the same basis (we shall consider here only von Neumann measurements), or equivalently in the same direc- tion on the Bloch sphere. After the measurement apparatus, two detectors count the number of particles, n+ andn− in each output mode (see figure 1). If the detectors are perfectly efficient (η= 1), one hasM =n++n−.
A. Testing the CH inequality
Our goal is to test a Bell inequality. Here we shall focus on the simplest Bell inequality, the CHSH inequality [11], which involves two inputs on Alice and Bob’s sidesA1, A2 andB1, B2, and two outputs, α, β ∈ {+,−}. For conve- nience we write it under the CH form [12],
CH =−P+A(A1)−P+B(B1) +P++(A1, B1) +P++(A1, B2) +P++(A2, B1)−P++(A2, B2)≤0 (2) whereP++(Ai, Bj)is the probability for both Alice and Bob to output ”+” when performing measurementsAiandBj re- spectively. Recall that under the hypothesis of no-signaling both CH and CHSH inequalities are equivalent [13]. Now, in order to test inequality (2), Alice and Bob must transform their data, basicallyn+ andn−, into a binary result, “+” or “−”.
A natural way of doing it is by invoking a voting procedure, for instance:
1.Majority voting: ifn+≥n− →“+”, otherwise “−” 2.Unanimous voting: ifn+=M →“+”, otherwise “−”
(3) or any intermediate possibility, for instance 2/3 or 3/4 ma- jority. For each voting method and given M corresponds a
thresholdN =dM/2e,d2M/3e, ..., Msuch that the outcome is+iffn+≥N.
At this point the two relevant questions are : first, is it pos- sible to violate the CH inequality with any of these voting procedures? Second, if yes, which strategy yields the largest violation? To address these questions one must compute the joint and marginal probabilities entering the CH inequality for each procedure.
B. Pure states
Let us first consider the pure entangled states
|ψi= cosθ|00i+ sinθ|11i (4) so that in equation (1), ρ = |ψihψ|. We will also write
|ΨMi = |ψi⊗M. For detectors with a perfect efficiency η = 1, all M particles are detected both on Alice’s and on Bob’s side. The marginal and joint probabilities entering the CH inequality for a vote with given thresholdN are
P+(A) =
M
X
n+=N
M n+
p+(A)n+p−(A)M−n+ (5)
P++(A, B) =M! X
nαβ∈Ξ
Y
α,β
pαβ(A, B)nαβ
nαβ! (6) wherep++(Ai, Bj) =tr(Ai⊗Bj|ψihψ|)andp+(Ai/Bj) = tr(Ai⊗11/11⊗Bj|ψihψ|)are the quantum joint and marginal probabilities for a single pair. Alice and Bob’s outputs are noted α, β ∈ {+,−}, nαβ is the number of pairs which gave detectionsαandβ, and Ξ = {nαβ ∈ N+|Pnαβ = M, nA+ =n+++n+− ≥N, nB+ =n+++n−+≥N}is the set of all events yielding the result “++” after voting.
Next one can choose the state |ψi and the measured set- tings. For the maximally entangled state of two qubits (θ = π/4) one may choose the standard optimal (for the caseM = 1) settings for the CH inequality, i.e. A0 = σz,A1 = σx, B0 = σx√+σz
2 ,B1 = −σ√x+σz
2 . Doing so with majority vot- ing (N =dM/2e), the CH inequality can be violated for any value ofM; the maximal amount of violation is numerically found to decrease with the number of emitted pairs asM−1.
Remarkably, a higher violation is found for different mea- surement settings, given by
A0=σz
A1= sin 2ασx+ cos 2ασz B0= sinασx+ cosασz
B1=−sinασx+ cosασz.
(7)
Withα ∼ 2√π2M−1/2, those settings are numerically found to be optimal. In this case the decrease ofCHis onlyM−1/2 (see figure 2(a)). The state leading to the largest violation is always the maximally entangled one (θ =π/4) for majority voting.
(a)
(b)
FIG. 2: (Color online) CH violation and resistance to noise for a source producingMindependent pairs. The states and settings used are discussed in the text.
(a) Maximal CH values for various thresholds : majority voting (full red line), 3/4 voting (dotted green line) and unanimity (dashed blue line). The decrease is asM−1/2for the first two, and exponential for the last one. The highest violation is thus reached using a majority vote.
(b) Resistance to noise for the different thresholds (same colors).
All curves decrease asM−1. The most resistant violation is that achieved by using majority voting.
The one-parameter planar settings (7) were already used by several authors [6, 7], for example in Bell experiments using a state we shall look at later (10) with the unanimous vote for anyM.
Performing numerical optimizations, we also found that a violation can be obtained for any voting strategy with any number of emitted pairsM (see figure 2(a)). We optimized the state (θ) and the four measurement settings, each time find- ing optimal settings of the form (7). For the unanimous vote, for instance, the optimal state is less and less entangled as the number of emitted pairsM increases, as described in [14], and the violation decreases exponentially withM. Thus for pure entangled 2-qubit states, the largest amount of violation is obtained with majority voting.
C. Resistance to noise
We now compute the resistance to noise that these viola- tions could bear, which is the relevant measure of non-locality considering experimental perspectives — the amount of viola- tion being basically just a number, without much significance in the present case as we shall see.
In a practical EPR experiment, imperfect detectors, noisy sources or disturbing channels introduce noise in the measure- ment results. To first order, this noise can be modeled at the level of the source, supposing that the produced pairs are not in the pure state|ψihψ|, but instead in a Werner state of the form
ρ=w|ψihψ|+ (1−w)11
4. (8)
The resistance to noise of a given violation is then defined by the maximal amount = 1−wof white noise that can be added to the pure state|ψihψ|such that the resulting stateρ still violates a Bell inequality (CH here).
Considering now that the sources of figure 1 produce the state (8), we look for the largest value ofwhich still gives a positive value ofCH, using settings of the form (7) and optimizing on the state (θ). For all voting strategies we find a resistance to noise decreasing like ∼ M−1, the major- ity voting being still the best choice (see Fig. 2(b)). Unlike when maximizingCHin the absence of noise, here the opti- mal state is always the maximally entangled one (θ =π/4), even for intermediary voting strategies, for which theCH vi- olation with this state decreases exponentially withM. This shows that appropriate figures of merit need to be used when examining practical situations.
These results are encouraging, but just as detectors might not be perfect, maybe the source cannot guarantee an exact number of pairsM, as needed here. To show that these vio- lations are relatively robust towards this issue, we now look at the case of sources producing a number of entangled pairs which follow a poissonian distribution.
D. Poisson sources
A poissonian source produces a stateρM ofM pairs with a poissonian probability
p(M) =e−µµM
M! (9)
whereµis the mean number of photon pairs. With such a source, a different number of pairs is created every time. So for a chosen voting assignment (3) the thresholdNvaries with the total number of photons detectedM =n++n−(we still consider perfect detectors), according to each realization.
Using settings of the form (7), we optimized numerically αand the state (θ) for several votes, in a situation where the source is poissonian. Doing so in order to get the largestCH violation and the highest resistance to noise, we obtained re- sults very similar to that of the fixedM case, verifying in par- ticular a decrease ofCHasµ−1/2for the majority vote and of
4
(a)
(b)
FIG. 3: (Color online) Maximal Bell violation and resistance to noise with a poissonian source of independent pairs. The red lines repre- sents the majority voting, the green dotted lines the 3/4 voting and the blue dashed lines the unanimity voting. Settings are chose in the form (7), optimal states are discussed in the text.
(a) For a large mean photon number, the decrease ofCH goes like µ−1/2for the majority and 3/4 vote, just like for the fixedM case.
Concerning the unanimity vote,CHdecreases faster than a polyno- mial, but slower than an exponential.
(b) Resistance to noise is very similar for all strategies, decreasing as µ−1just like with as source of fixed pairs number.
the resistance to noise asµ−1(see fig. 3). Similarly, the states yielding the largestCH values are the maximally entangled one for the majority vote, and partially entangled ones for the two other votes. A difference, however, is thatCH is found to decrease slower than exponentially for the unanimity vote.
Note also that since it is possible to findCH >0, and the probability to get a+result vanishes forµ → 0, there exist an optimalµ= 1.2∼1.8yielding a maximum CH violation.
But this feature is not found in the resistance to noise.
E. Inefficient detectors
We now consider detectors with finite efficiency η < 1, and look in what circumstance a Bell violation can still be
FIG. 4: (Color online)CHviolation with inefficient detectors, as a function of the probability for a photon to be detected. The upper thin curve shows the traditional caseM = 1with known critical efficiencyη= 2/3[15]. The two other curves are forM = 5pairs with majority (full red line) and unanimity voting (dashed blue line).
observed in a multipair scheme with such detectors.ηis to be understood here as the probability for a particle to be detected.
In general, in presence of detectors inefficiencies (or parti- cle losses) the total number of particles detected by Alice and Bob are different (nA++nA− 6=nB++nB−). Thus, for a given voting strategy, the thresholdsN applied by Alice and Bob might be different for the same event, since it depends on the total number of photons detected by each party. Testing a Bell inequality in this situation without appealing to post-selection introduces no detection loophole, but it is not a surprise to find that high efficiencies are needed in order to find a Bell violation in such circumstances. Figure 4 shows the maximal CH violations obtained (optimizing on states and settings) as a function of the detection efficiency forM = 1andM = 5 with majority and unanimity voting. The required detector ef- ficiency increases with the number of pairs, leaving no chance to find a violation at highM withη <<1.
One way to deal with detector inefficiencies consists in post-selecting events in which exactlyM photons are detected on both sides. This way cases in which particles were not de- tected are neglected and the Bell violation is recovered inde- pendently of the losses. (If detectors also have dark counts, noise will appear in the statistics, which can be treated with Werner states as presented in the ’resistance to noise’ section.) This approach is however not perfect as it is subject to the so-called detection loophole [16] : there exist local models, exploiting detectors inefficiencies, that can violate a Bell in- equality [17]. But also, one needs to know exactly the number of pairsM created, before measuring them. This last con- dition might not be guaranteed, for example with poissonian sources where the knowledge ofM is often inferred from the number of detected particles.
To estimate the impact of losses, we consider the case in which exactly 1 of theM photons flying to Alice, and 1 go- ing to Bob, are not detected. As the number of created pairs increases, this is a situation that must happen frequently even with very efficient detectors. Using the majority and unanim-
ity vote in this situation we numerically verified that theCH inequality couldn’t be violated, at least forM ≤50.
A way to understand this result is by noting that the sets of events yielding results + and - are separated by only 1 photon number. Thus, removing one photon mixes the two sets. It should thus be advantageous to separate these two cases such that for instancen+ ≥N →+,n− ≥N → −,M −N <
n+ < N →∅. Using this particular post-selection we could find a Bell violations in the case of 1 photon loss on both sides, withN =M−1(unanimity voting), starting atM = 5. For details on this post-selection, see ref [14].
III. INDISTINGUISHABLE PHOTONS
In the first part of this work we showed how, using multiple independent pairs together with independent global measure- ment on all the photons produced, one could find a substantial CH violation, even in the presence of lots of pairs. But how good is this compared to a source producing the2M photons altogether? For the sake of comparison we now consider a specific example, commonly produced in many labs. By the same occasion it will uncover some aspects of the relation be- tween entanglement and non-locality.
The state we are discussing now can be written asρM =
|ΦMihΦM|with
|ΦMi= 1 M!√
M+ 1
a†0b†0+a†1b†1M
|0i (10) wherea0,a1 are orthogonal modes on Alice’s side and b0, b1 orthogonal modes on Bob’s side (for instance horizontal and vertical polarization modes). A way to produce this state is with a parametric down conversion (PDC) source, which gives a poissonian distribution of such states. The same global measurements as previously performed onM photons can be realized here by just using the same setup as before : a po- larizer followed by two photon counters on each side (same setup as represented in Figure 1, but with a different source).
Considering state (10) we make a similar analysis as previ- ously, briefly reviewing the results of [6, 7] for the amount of violation achievable, and presenting our own analysis for the resistance to noise.
We computed the new probabilities entering the CH ex- pression for this specific state and, choosing various voting procedures, numerically optimized the settings according to α(7) in order to get the largestCH violation. Surprisingly, for any number of photonsM, all voting procedures yield ap- proximately the same maximum violation ofCH, decreasing asM−1(see figure 5(a)). This is even more surprising as the settings needed for that are not the same for all voting meth- ods.
Note that Reid et al. [6] used another figure of merit : S = CH+BB (withB = P+A(a1) +P+B(b1)), which gives different results for the different voting strategies. Recalling the artefacts we already found in the amount of CH violation for poissonian sources, we choose to look now at an experi- mentally meaningful figure of merit, namely the resistance to noise.
(a)
(b)
FIG. 5: (Color online) Comparison between sources producing inde- pendent pairs or indistinguishable photons, using settings of the form (7).
(a) Maximal CH violation achieved with a source of indistinguish- able photons for various voting procedure (superposed black dots).
Compared to the previously calculated violations for independent pairs (same curves as in figure 2(a)), unanimity voting (lower blue dashed line) yields less violation, while majority voting (upper red line) yield the highestCH values. Note that the maximal violation with indistinguishable photons almost doesn’t depend on the voting procedure used.
(b) Maximal resistance to noise in the majority voting scenario (full red lines) and the unanimity scenario (dashed blue lines) for sources producing independent pairs (thick line, same curves as in figure 2(b)) or indistinguishable photons (thin line). The unanimous vote is more robust for indistinguishable photons, but majority voting on independently produced pairs yields the most persistent violation.
A. Noisy symmetric state
Unlike for distinguishable photons, the effect of a noise map on a symmetric M-photons state does not affect each photon independently. We thus need here a more precise noise model. For the sake of simplicity, we put ourselves in an asymmetric setting, modeling the noise observed in the state measured by Alice and Bob as coming from the imperfection of the channel linking the source and Alice. Because the chan- nel slightly deteriorates the systems passing through it but has
6 no preferred basis, we model it by an average over all rotation
axes~n= (sinθcosφ,sinθsinφ,cosθ)in the Bloch sphere of rotationsU by an angle2β, withβfollowing a properly nor- malized gaussian distributionp(β) = 2
(1−e−2σ2)√2πσe−β
2 2σ2. For any representation~σ= (J~x, ~Jy, ~Jz)of SU(2) generators, the rotation operator isU = exp (−β~n·~σ). The state after the noisy channel is thus given by :
ρout = Z
SU(2)
p(β)(U⊗11)ρin(U†⊗11)[dU] (11)
= 1 4π
Z ∞
−∞
dβ Z π
0
dθ Z 2π
0
dφsin2βsinθp(β)(12)
×(U(β, θ, φ)⊗11)ρin(U†(β, θ, φ)⊗11) (13) where we have used the appropriate Haar measure of SU(2) in terms of the Euler angles :[dU] = 4π1 sin2(β) sin(θ)dβdθdφ.
We introduced the Haar measure here because it is the only measure which is invariant under group operations. It thus treats every rotation the same way, reflecting the fact that the noise has no preferred rotation axis.
This channel applied to a single pair stateρin=|Φ1ihΦ1| with|Φ1i = √1
2(a†0b†0+a†1b†1)|0i produces a Werner state (8), allowing one to make a correspondence between the usual noise model used in the previous part of this work, in terms of Werner states and this one :
3w=e−2σ2+e−4σ2+e−6σ2. (14) This relation allows us to interpret the amount of white noise = 1−was being, to first order, the variance of the random rotation angle :
= 4σ2+O(σ4). (15) Applying this channel to the state|ΦMifor variousM, and performing the majority and unanimity votes with settings in α(7), we found that the unanimity procedure is more robust to noise than the majority vote, scaling like∼M−1(see figure 5(b)).
B. Particle losses
To compare indistinguishable and independent pairs in the case of losses, we consider the case in which one particle is lost on each side, yielding a total number of detections2(M− 1). In terms of modes, the state measured after the loss of particles can be written
ρM−1∼a0b0|ΦMihΦM|a†0b†0+a0b1|ΦMihΦM|a†0b†1 +a1b0|ΦMihΦM|a†1b†0+a1b1|ΦMihΦM|a†1b†1.
(16) Using such a state, we could find a violation for sufficiently many pairs forM ≥10, starting majority voting. Thus, there is no need for additional post-selection here.
IV. DISTINGUISHABLE VS. INDISTINGUISHABLE PAIRS In the last sections we examined how two different mul- tiparticle bipartite states|ΨMi(4) and |ΦMi (10) could be used to show nonlocality using a natural setup producing bi- nary outcomes. These two states are actually related : if one were to produce the state of independent pairs|ΨMiwith fun- damentally indistinguishable photons on both Alice and Bob’s sides, then the state created would be symmetric with respect to permutations between Alice’s photons or Bob’s ones, and we would actually have produced state|ΦMi. This can be seen by projecting|ΨMionto the corresponding symmetric subspaces :
|ΨMi=
|0iA|0iB+|1iA|1iB
√2
⊗M
=2−M2
M
X
i=0
X
π∈ΠMi
π|0i⊗iA|1i⊗M−iA ⊗π|0i⊗iB |1i⊗MB −i
−sym−−→2−M2
M
X
i=0
|i, M−iiA|i, M−iiB∼ |ΦMi (17) whereΠMi is the set of all Mi
possible arrangements of i
“0” andM−i“1”, and|i, M −iiA= (a
†
0)i(a†1)M−i
√i!(M−i!) |0iis the Fock state describingiof Alice’sM photons in the “0” state andM−iin the “1” state.
So the only difference between|ΨMiand|ΦMiis the dis- tinguishability of theM photons flying to Alice or Bob. But in the setup we considered (as described in figure 1), we didn’t take advantage of the particular pairing between some of Al- ice’s photons with some of Bob’s ones. Because we applied global measurement, we could even suppose that all photons on Alice(Bob) side were mixed before reaching the beamsplit- ter. In other words we classically lost trace of the pairing be- tween Alice’s and Bob’s photons. We are thus comparing a situation in which one explicitly chose not to distinguish be- tween photons belonging to a given set, with another one for which these photons are intrinsically indistinguishable.
Let us now compare the entanglement present in both states.
Eisert et al.[5] calculated the amount of entanglement present in the state of distinguishable particles|ΨMiafter having for- gotten the pairing of Alice’s photons with Bob’s ones. ForM even :
Ed=E(|ΨMi) =
M/2
X
j=0
(2j+ 1)2 2M(M+ 1)
M+ 1 M/2−j
log2(2j+1).
(18) Concerning the state of indistinguishable particles|ΦMi, writ- ing it in terms of modes as in equation (17) we see that its entanglement is given by
Ei=E(|ΦMi) = log2(M + 1) (19) since it is a maximally entangled state of two systems of di- mensionM. Evaluating these two quantities, we findEi >
Ed ∀M, and more preciselyEi/Ed
−−−−→M→∞ 2. So more en- tanglement is present in the state where photons are quan- tumly indistinguishable, but a larger violation of CH can be observed using a natural setup if the photons are in princi- ple distinguishable, but we choose not to make any difference between them. Looking at how resistant these violations are with respect to noise confirms this order. Only should it be noted that compared to particle losses, the indistinguishable case looks more resistant, since no additional post-selection was necessary to find a violation when both Alice and Bob lost a particle during the experiment.
This is in agreement with other results [18], showing that entanglement and nonlocality are different measures.
V. EXPERIMENTAL PERSPECTIVES
In this Section we give a brief overview of experimental situation where our techniques might be applied.
As mentioned previously, the experiment of Ref. [3] shows evidences for entanglement in ensembles of ultracold atoms of
87Rb in an optical lattice. Entanglement between two atomic levels is generated via a partial swap gate, an entangling op- eration. In order to apply our techniques, the atoms of each level should be addressed separately; that is Alice should hold all atoms in the ground state, and Bob all atoms in the excited state. Note that in this experiment the pairs are independent because they are located in different regions of the optical lat- tice.
Another experimental situation invoking Bose-Einstein condensates where our techniques might be useful is super- radiant scattering [19]. It has been argued that this process generates entanglement between the emitted photons and the atoms of the condensates. In that case the particles would be indistinguishable.
A third possibility is the experiment discussed in Ref. [20], which is a proposal for energy-time entanglement of quasipar- ticles in a solid-state device. This experiment is the adaptation of the Franson-type experiment [21] with entangled electron- hole pairs.
Finally it is also worth mentioning quantum optics. How- ever it is not clear that our techniques will turn out useful in this field, since they require high detection efficiencies, a feature that still lacks generally in optics. Still, sources pro- ducing independent entangled pairs, or indistinguishable pho- tons via parametric down-conversion (PDC), are already well understood. Multi-photon entanglement using PDC sources was demonstrated in [10]. A careful analysis of post-selection might thus open the possibility to feasible experiments.
VI. CONCLUSION
We considered Bell experiments on multiple pairs of parti- cles, where the two parties are not able to address each particle separately and thus call upon global measurements, projecting all of their incoming particles in the same basis. Votes were introduced as a natural way to produce binary outcomes from
TABLE I: Summary of the main results of this work.
two detection numbers. This allowed us to test the CH in- equality in the presence of both a source ofM independent pairs and ofM indistinguishable pairs, highlighting a viola- tion of the CH inequality for any number of pairsM. Con- sidering the resistance to noise of such violations, modeled as a noisy channel, we could provide an experimentally mean- ingful measure of nonlocality. The impact of losses was also evaluated for the two situations, showing that indistinguish- able pairs are more robust against losses. More detailed re- sults are summarized in table I. Finally, a comparison of the nonlocality observed for each source with the entanglement of their respective states provided another example of non- monotonicity between these two quantities.
VII. ACKNOWLEDGEMENTS
We thank F.S. Cataliotti for pointing out the potential appli- cation of our results to superradiant scattering.
We acknowledge financial support from the EU project QAP (IST-FET FP6-015848) and the Swiss NCCR Quantum Photonics.
[1] J. Bell, Physics1, 195 (1964).
[2] A. Aspect, Nature398, 189 (1999).
[3] M. Anderliniet al., Nature448, 452 (2007).
[4] Y.-C. Liang and A. C. Doherty, Phys. Rev. A73, 052116 (2006).
[5] J. Eisert, T. Felbinger, P. Papadopoulos, M. B. Plenio, and M.
Wilkens, Phys. Rev. Lett.84, 1611 (2000).
[6] M. D. Reid, W. J. Munro, and F. De Martini, Phys. Rev. A66, 033801 (2002).
[7] P. D. Drummond, Phys. Rev. Lett.50, 1407 (1983).
[8] S. J. Jones, H. M. Wiseman, and D. T. Pope, Phys. Rev. A72,
8 022330 (2005).
[9] G. Toth, C. Knapp, O. Guhne, and H. J. Briegel, Phys. Rev.
Lett.99, 250405 (2007).
[10] H. S. Eisenberg, G. Khoury, G. A. Durkin, C. Simon, and D.
Bouwmeester, Phys. Rev. Lett.93, 193901 (2004).
[11] J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Phys.
Rev. Lett.23, 880 (1969).
[12] J. F. Clauser and M. A. Horne, Phys. Rev. D10, 526 (1974).
[13] D. Collins, and N. Gisin, J. Phys. A : Math. Gen.37, 1775 (2004).
[14] N. Brunner, C. Branciard, and N. Gisin, arXiv:0802.0472.
[15] P. H. Eberhard, Phys. Rev. A47, R747 (1993).
[16] P. Pearle, Phys. Rev. D2. 1418 (1970).
[17] N. Gisin, and B. Gisin, Phys. Lett.A260, 323-327 (1999).
[18] A. A. M´ethot, and V. Scarani, Quantum Inf. and Comput.7, 157 (2007).
[19] S. Inouyeet al., Science285, 5427, 571 (1999).
[20] V. Scarani, N. Gisin, and S. Popescu, Phys. Rev. Lett. 92, 167901 (2004).
[21] J. D. Franson, Phys. Rev. Lett.62, 2205-2208 (1989).
